(26) Shilling, I V . G., Laxton, A. E., Phil. Mag. 10,721 (1930). (27) Srivastava, €3. N., Saxena, S. P m . Phys. 5 0 6 . B70, 369 (1957) ; J . Chem. Phys. 27, 583 (1957). (28) Trautz, M., Binkele, J. E., Ann. Physik 5 , 561 (1930).
c.,
(29) Trautz, M., Kipphan, K. F., Zbid., 2, 746 (1929). (30) Trautz, M., others, Zbid., 2, 733 (1929); 5, 561 (1930); 7, 427 (1930); 20, 118, 121 (1934). (311 Vasilesco. V.. Ann. bhvs. 20. 292 (1945). (32) Walker, R. E., \V;,s;enberg, A.'A., 3. Chem. Phys. 29, 1139, 1147 (1958); 31, 519 (1959); 32, 436 (1960).
(33) Weissman, S.: Mason, E. A,, Physica 26, 531 (1960). (34) Weissman: S., Saxena, S. C., Mason, E. A,, Phys. Fluids 3, 510 (1960). (35) Ihid., 4, 643 (1961). (36) Wilke, C. R., J . Chem. Phys. 18, 517 (1950). (37) Wobser, R., Muller, F., Kolloid Beih. 52, 165 (1941). RECEIVED for review July 3, 1961 ACCEPTEDMay 7, 1962
RATE EQUATIONS FOR CONSECUTIVE HETEROGENEOUS PROCESSES KENNETH 6 . BISCHOFF1 AND GILBERT F. FROMENT Lahoratorium voor Organische Technische Chemie, Rijksuniversiteit te Gent, Gent, Belgium
A general discussion of the forms of rate equations for consecutive, heterogeneous processes is given.
The complexity oil the derivation of these rate equations depends on whether or not the process has parallel steps combined with the consecutive ones. The procedures that can b e used for cases with only consecutive steps are outlined. The particular case of a heterogeneous catalytic reaction with more than one ratecontrolling step is covered in detail. It may be extremely difficult to determine if more than one step is controlling. However, kinetic constants derived from experimental data on the basis of only one step controlling may be greatly in error.
N hiAw
heterogeneous chemical reaction systems of engi-
I neering interest, the chemical reaction is only one of several
consecutive steps. If the rate coefficients of all of the other steps are extremely large lvith respect to that of the reaction itself (in other words, if their resistances are very small), the rate of the over-all process may be obtained by inserting the over-all potential difference in the rate equation for the chemical reaction. In general? however, because of the influence of the other steps, the potential difference to be used in the chemical rate equation is some unknown fraction of the over-all potential difference, the only one that can be measured. However, since at steady state the rates of each of the steps must be equal, a set of equations may be written between which the intermediaize potentials can be eliminated by algebraic manipulation. This yields an expression that contains only the over-all potential difference and a coefficient that combines the rate coefficients or resistances of the different steps. T o chemical engineers. the most familiar applications of this method of eliminating intermediate, unknown potentials are the treatment of simple heat conduction through a series of la)-ers with different th-rmal conductivities and the t\ro-film theory of mass transfer. These are examples where the relations between rates and potential differences are linear. The method has been less intensively applied to chemical reaction systems, probably because chemical reaction rates are often dependent in a nonlinear way on the driving force, although this should not be considered as a restriction. I n addition, however, heterogeneous reaction systems are often not truly consecutive, and also contain parallel steps, which greatly complicate the treatment. This is the case when the phase in ivhich the reaction takes place is permeable-for instance, in fluid-porous solid systems or in fluid-fluid systems. I n nonpermeable reaction phases on the other hand-for instance, fluid-nonporous solid systems with reaction a t the solid surface
-only consecutive processes occur. This paper is concerned only with the nonpermeable reaction phase case-i.e., with truly consecutive processes. Method of Combination of Resistances
Linear Processes. The first case to be considered is that of a first-order reaction between one component of a gaseous mixture and a solid. The reaction at the solid surface will be in series with mass transfer through the fluid. The rate of mass transfer is expressed by kD(C
ID
- C,)
(1 1
while the chemical reaction rate is given by
(2)
r e = k&t
At steady state the two rates are equal, so that Y
= ~ D ( C- ci) =
kec,
from which c, may be found to be:
M'hen Equation 3 is substituted into Equation 1 or Equation 2> - c =
I
- A kD '
1
k,c
(4)
where
k, is the "over-all" rate coefficient, taking both reaction and mass transfer into account. Equation 4 is well known (3-5) and is of exactly the same form as the equations used in the two-film theory. When k , + m , the chemical reaction is said to be rate-controlling., and Equation 4 becomes
Present address, Univtrsity of Texas, Austin. Ter.
(6)
r = k,c VOL.
1
NO. 3
AUGUST 1962
195
This merely means that the interfacial and bulk fluid concentrations are the same. When k, + 03, mass transfer is controlling, and it follows from Equation 4 that r = kDc
(7)
which means that the interfacial concentration of the reactant is zero. Extension to Nonlinear Processes. The same procedure may be used for a second-order chemical reaction in series with mass transfer. The rate equations then are I D = kD(C
- Ci)
rc = k,ci2
Upon elimination of c i from Equations 8 and 9,
For k ,
+ 03
and k,
-
03,
respectively, Equation 10 reduces to r = k 2
r = knc
Thus Equation 10 is the combined rate equation, similar to Equation 4. In this case, however, the concept of a n “overall” rate coefficient has no meaning, since the functional forms of the driving forces in Equations 8 and 9 are different. I t is true that Equation 10 could be written in the form r = ko*C
(13)
where
would be a so-called “effective” mass transfer coefficient. Equation 13 then treats the situation as mass transfer corrected for chemical reaction. It is seen from Equation 14 that k , is not independent of the bulk fluid concentration, G, and thus is generally not constant. Therefore it would seem preferable to use Equation 10 itself and not to try to think of the process as a “corrected” mass transfer process, even though the latter picture is more familiar. If the same procedure is used for an nth-order reaction, the lollowing result is obtained :
Equation 15 cannot be solved explicitly for r for arbitrary n. Therefore no expression for an over-all or effective coefficient such as Equation 5 or Equation 4 can be obtained. Instead, the values of r = r(k,,k,,c) may be found numerically from Equation 15. In general, for several consecutive reactions,
The unknown concentration in Equation 16 would have to be eliminated numerically. This type of procedure has not been appealing for calculation purposes, probably for two reasons : I t is complicated, and it does not lead to a n explicit, easy to use, familiar equation such as Equation 4, obtained when the chemical reaction is first order. Therefore, calculations have usually been limited to the cases where only one of the resistances is controlling. However, with the use of high speed computing machines these objections are no longer serious and should not restrain the use of the generalized procedure of combination of resistances. 196
l&EC FUNDAMENTALS
It follows that the use of the general equations for design purposes presents no serious difficulties. The reverse problem of the evaluation of the various coefficients from experimental data is much more complex, however. I n heat or mass transfer studies, the forms of the rate equations are known and only the coefficients have to be found from the experimental data. I n kinetics studies, on the other hand, the form of the rate equation itself is unknown and this complicates the problem severalfold. Because of the large number of possible forms of the rate equations, the testing of each of them in combination with the other steps is an almost impossible task. What is sometimes tried is to arrange experimental conditions such that only one of the steps is controlling for any one experiment, so that the various coefficients may be evaluated by separate experiments. In most cases, it is not possible to do this, and then the more complicated procedures involving combination of resistances must be used in interpreting the experimental data. We discuss below a n example that shows the ease with which experimental data can be misinterpreted when all resistances are not taken into account in the evaluation of rate coefficients. Rate Equations for Consecutive Nonlinear Catalytic Processes
The Langmuir-Hinshelwood adsorption-surface reaction mechanisms, as applied by Hougen and Watson ( 6 ) ,are widely used in chemical engineering kinetics. Three consecutive steps occur when a fluid reacts on a catalytic surface (not counting fluid phase diffusion) : adsorption of reactants, surface reaction, and desorption of products. Each step may be complicated by such effects as dissociation of adsorbed compounds-adsorption of only one reactant or all reactants, for example-and an extremely large number of combinations between them is possible. For this reason, it has always been assumed until now that one of the steps was definitely controlling. and it was considered that this was a sufficiently accurate approximation. Even with this assumption, there are still many possible mechanisms-for example, 17 different mechanisms were tested in the hydrogenation of iso-octenes ( 6 ) . Recently, however, Thaller and Thodos ( 8 ) have reported data on a reaction that seemed to show two different mechanisms controlling, depending on the temperature levels. At low temperatures their data indicated surface reaction controlling, while at high temperatures the desorption of one of the products seemed to be controlling. Obviously. then. in the range between those extremes the rate of the process is determined by a combination of the rates of surface reaction and desorption, and the assumption of one controlling step does not hold. In the following, the method of combination of resistances is applied to the same series of processes as those involved in Thaller and Thodos’ work. I t is shown as a result of this study that examples of more than one controlling step might be fairly widespread, but have not been previously noticed because of lack of investigation of a wide enough temperature range. The over-all reaction is A=S+H
(17)
and is considered to consist of the following steps. (18)
A+I$Al
AI
+1
S1
+ HI
(19)
sl=s+l
(20)
HIgH+I
(21)
Both Reactions 19 and 21 will be considered to be controlling.
least squares with only integral data (2, 9). and more sophisticated methods are required. Yang and Hougen ( 1 7 ) have shown that use of initial rate data is a very convenient way to do this. However, even initial rate data are not always sufficient to determine the mechanism. T h e expression for initial rates when both desorption and surface reaction are controlling may be found from Equation 32,
Then Reactions 18 and 20 attain equilibrium, so that =
6.41
GSI
KApAcci
= Kspscl
T h e rate of the surface Reaction 19 is
and the rate of desorption of H is YH = kH/(CHI
- KHPACI)
Substituting Equations 22 and 23 into Equation 24 gives
If only one of the steps is rate controlling, Equation 33 reduces, of course, to the proper expression for that case as given by Yang and Hougen ( 7 7). For k , + m the desorption is controlling, and
.
Substituting Equations 22 and 23 into Equation 27 gives, after rearranging,
(34)
In0 = kH
For kH
+
mI
the surface reaction is controlling and
Now, Equation 28 is substituted into Equations 25 and 26, Establishment of Rate Equations for Consecutive Processes from Experimental Data
T h e only unknown left in Equations 29 and 30 is cHI. .4t steady state, y H = r K = Y , thus cHI may be found by equating Equations 29 and 30. If this is done, and the expression for cHI is substituted back into either Equation 29 or 30, the result is
T h e initial rates can now be compared for various situations. Figure 1,a and b. shows the initial rate curves as functions of pressure for various values of kR kH. The value of kH = 0.122 was chosen so that when desorption was controlling (r0 = ka = 0.122) the rate was the same as the mean rate when surface
--
r = -KskR
I
kR:10
Kn
02-
01
where k , = rtkR’ and kH = G t k H ’ . If Equation 31 is multiplied out and solved for r, r
where
(P. =-__--
fly)
2A‘ ________B’ - d/B%qm
K
=
KAKR/KsKH
(32)
0
5
10
15
10
15
KAPA
(32a)
I
0.2
kH: 0,122
I
Equation 32 has been written in the form of the over-all potential difference divided by a total resistance. It is rather complicated, however. and it cannot be written as a linear equation in the pressures, pa. p s , and p H . Therefore it would be impossible to use integral reactor data with a least squares fit to determine the constants here. This is not too serious, however, since it is more or less generally accepted today that the proper mechanism cannot be determined by the use of
0
5 KA
Pn
Figure 1. Initial rate curves for combined surface-reaction and desorption-controlling mechanism a. Approaching pure surface-reaction controlling b. Approaching pure desorption controlling VOL. 1
NO. 3
AUGUST 1962
197
reaction vias controlling with kR = 1.0
15
0.3
Figure l,a, shows curves that are approaching pure surfacereaction controlling. The curves all have the same shape. In other words, by inspection alone, one cannot tell whether the initial rate curves result from the combined surfacereaction and desorption controlling or from the pure surfacereaction controlling mechanisms. Figure 1,b, shows curves that are approaching pure desorption controlling. Until the pure desorption line is approximately reached, these curves also look similar. Therefore, by only qualitative visual inspection, it would be difficult to tell how closely the ratio k,/k, approaches k, k, = a (pure desorption). Figure 1,a and b. shows that further information than the pressure dependency of initial rates may be needed to determine the mechanisms correctly. I t is also clear from Figure 1,a and b , that the combined rates are always lower than either of the individual ones. This means that the procedure of calculating the rate based on one
02
I
--'\ '0
0.1
\ . -
--
0
o
5
0
Figure 2.
550
15
10
P
600
- a1m.
Initial rates calculated from Equation 33
2!
T =6OO'K
60
5 50 20
50
LO
15 LUI LO
> 650
30
10
700
20
750
5
803
(
IO
3
0
5
10
15
-
P atm. A
Figure 3. Plot for determination of coefficients from experimental data assuming surface-reaction controlling 198
l&EC FUNDAMENTALS
mechanism and then on the other, and choosing the lowest, is correct only when one of the rates is extremely large with respect to the other. If this is not so, the complete equation combining the different resistances must be used in order to calculate the proper rates. T o show more clearly the difficulties involved in interpreting data where several steps determine the rate, the following experimental program was simulated. A set of typical values of catalytic rate coefficients was used in the combined rate equation (Equation 33) to generate a set of “experimental” data. These data were then submitted to a kinetic analysis, to determine rate coefficients based on the (false) assumption that surface reaction alone was controlling. The various coefficients used were taken at 600’ K. to be k~
=
IIH = 0.5
0.5
k~
=
kR,1,+,Ka : T r u e Coefficients hR,KA.from “ d a t a ”
1.0 atm.-’
T o get “data” a t different temperature levels, the following activation energies were used: ER
=
17 kcal./gram-mole
EE
=
9 kcal./gram-mole
E A = -5 kcal./gram-mole
T h e coefficients are given in a n Arrhenius plot (Figure 4). Using these coefficients, initial rate curves a t each temperature were calculated with Equation 33, and are given in Figure 2. From this point on, ign’orancewill be assumed of the fact that the curves of Figure 2 were actually calculated on the basis of a combined desorption-surface reaction rate equation. By a visual comparison of the “data” in Figure 2 with the curves given by Yang and Hougen ( I I ) , the conclusion would be reached that the reaction might have a surface reacrioncontrolling mechanism. From Equation 35 it can be seen that a plot of
d$
us.
Pa should
12
13
14
15
16
17
IS
19
10oo/r
Figure 4.
Arrhenius plot of coefficients
then be a straight line, and
Figure 3 is such a plot (of the ”data.” T h e calculated points, of course, do not give a perfect straight line, but with scattered experimental data with =t5yO error as used on Figure 3, the difference from a straight line would not be noticed. Especially in view of the qualitative agreement of Figure 2 with Yang and Hougen’s curves for pure surface reaction-controlling, the investigator would be satisfied with the chosen pure surface-reaction mechanism. Continuing, the surface-reaction coefficients would be calculated from the slope and intercept of the straight lines of Figure 3. This was done with the data points by the use of a least squares fit of the data (Table I). The difference between the true values and the ones calculated on the basis of pure surface-reaction controlling is large. Thus, even though the data as shown in Figures 2 and 3 seem to follow a surfacereaction mechanism fairly closely. the final result, k,, the surface-reaction rate coeffic.ient>is very much in error. T h e next step of the investigator would be to make a n Arrhenius plot of the determined coefficients. These are shown on Figure 4>along with the true coefficients. The data points deviate slightly from a straight line, but, again with the assumed error of ? ~ 5 7 ~ the , difference would not be noticed. Therefore, with the results of Figures 2, 3, and 4, the investigator would come to the conclusion that the reaction has a pure surface reaction-controlling mechanism. IVithin the limits of the accuracy of experimental kinetic data, there would be no way of knowing that the coefficients determined are greatly in error. If the data are to be used for design purposes only, the situation is not really too serious. Even though the rate coefficients as determined are in error, if they are consistently
used in Equation 35 for the same ranges as the laboratory experimental data, the proper rates of reaction will be calculated. I n fact, as pointed out by Weller (70).for use in the scale-up of extensive data, the complicated adsorption equations are not really necessary, since other simpler. empirical equations may give just as good results. However, from the point of view of predicting reaction rates from basic data with the aim of ultimately eliminating the pilot plant stage, the adsorption mechanism concept is still about the only generally useful logical system for catalytic kinetics and will probably be used for many years ( I , 7). From this point of vie\\., the deviations shown in Table I are serious. Attempts are now being made to correlate rate coefficients of catalytic reactions as functions of the chemical properties of the components of the reaction systems ( 7 ) . For studies such as this, the deviations are particularly serious, because valuable correlations might be rejected merely because incorrect rate coefficients are used as starting information.
Table 1.
Coefficients
k R , Surface Reaction
kA, Adsorption Coeflcient
Coeficient Temp.,
K. 550 600 650 700 750 800
True 0,137 0.50 1.495 3.82 8.55 17.55
7%
From
deaia-
linta
tion 25 34 41 49 57 63
0.103 0.332 0.877 1.93 3.69 6.5
VOL. 1
True 1.462 1.0 0,725 0.550 0,433 0.350
From data 1.12 0.720 0.506 0,383 0.319 0,276
NO. 3 A U G U S T 1 9 6 2
70
deoza-
tion
23 28 30 30 26 21
199
S = component S S1 = adsorbed S
Conclusions
Obtaining precise kinetic data for catalytic reactions is an even more formidable task than has been previously thought. The need for the consideration of more than one rate-controlling step can be revealed only by extensive experimentation, covering extremely wide ranges of variables.
c
Acknowledgment
D i
SUBSCRIPTS
A, H? S, Al, H1, S1 = components A? H, S:‘41: H1, SI, respectively
I
= chemical reaction = mass transfer = at solid surface
= vacant active sites = over-all
One of the authors (K.B.B.) gratefully acknowledges assistance from the National Science Foundation (U.S.A.) in the form of a postdoctoral fellowship.
R
= surface reaction
t
= total active sites
Nomenclature
literature Cited
A = component A A‘ = defined by Equation 32b A1 = adsorbed A B‘ = defined by Equation 32c G = concentration, moles per cu. meter C’ = defined by Equation 32d E = activation energy, kcal./gram-mole H = component H HI = adsorbed H k = rate coefficient k* = effective mass transfer coefficient K = over-all equilibrium constant. or with subscripts, adsorption equilibrium constants 1 = active site n = order of reaction p = partial pressure, atm. r = rate
(1) Boudart, M., A.Z.Ch.E. Journal 2, 62 (1956). (2) Chou, C., IND.ENG.CHEM. 50, 799 (1958). (3) Frank-Kamenetzki, D. A., “Stoff- und Warmeiibertragung in
o
der chemischen Kinetik,” Springer Verlag, Berlin, 1959. (4) Froment, G., Znd. Chim. Belge 25, 245 (1960). (5) Hofmann. H.. Bill. W.. Chem. Iner. Tech. 31. 81 11959). (6) Hougen, 0.A., Watson. K. M., “Chemical Prockss P&iples,” Vol. 111, Wiley, New York, 1947. ( 7 ) ,Jungers, J. C., et al., “Cinttique chimique appliqute,” Technip, Paris, 1958. (8) Thaller, L. H., Thodos. G., A.Z.Ch.E. Journal 6, 369 (1960). ( 9 ) Walas. S. M.. “Reaction Kinetics for Chemical Encineers.” McGraw-Hill, New York, 1959. (10) Weller. S., A.I.Ch.E. Journal2, 59 (1956). (11) Yang. K. H., Hougen, 0. A.. Chem. Eng. Progr. 46, 146 >
,
v
(1950).
RECEIVED for review March 20, 1961 A C C E P T E D August 31, 1961
STABILITY OF ADIABATIC PACKED BED REACTORS. A N ELEMENTARY TREATMENT -
SH EA N L I N L I U A ND NEA L R
.
A M U N D S 0 N, University of .Winnesota, .Minneapolis 14, .Minn.
The problem of stability of a packed bed adiabatic catalytic reactor is considered for a simple model in which mass and heat transfer resistances are lumped at the particle surface and the only intraparticle effect is that of chemical reaction. As has been shown, such a catalytic particle may exist in more than one state. A fixed bed reactor may contain particles whose states are determined by their past histories. The transient equations for the reactor are written and solved by the method of characteristics. The solutions show that under certain conditions, amenable to a priori prediction, there will be nonunique temperature and concentration profiles. In fact, there may be a multiply infinite set of profiles, depending upon the initial state of the reactor. Calculations are made for a series of initial temperatures.
HE PURPOSE of this work was to consider the stability Tof adiabatic, fixed bed catalytic reactors. The stability of continuous, well agitated homogeneous reactors was examined by van Heerden (6) and others (2, 4). The latter problem is not difficult because the transient behavior is described by first-order ordinary differential equations. The packed bed reactor is much more difficult, even for the simplest nontrivial physical model, since the analogous mathematical justification is not available for partial differential equations. Barkelew (3) made an extensive numerical study for a packed bed system and arrived a t some empirical generalizations. Wagner (7) studied the
200
ILEC F U N D A M E N T A L S
stability of a single catalytic particle and showed the existence of multiple steady states in an analysis very similar to that used for stirred tanks. Cannon and Denbigh (5) examined the thermal stability of a single reacting particle. Wicke and Vortmeyer (8-7 7) considered the packed bed reactor, and the purpose of the work described here was to extend their results and examine the problem in somewhat more detail. Their treatment was in terms of the steady state, while this work considered the transient behavior and its consequences. A very simple model for a packed bed reactor is considered. Because it is adiabatic there is no radial transport of heat or mass, and, in addition, it is assumed that axial transport, save
